Independence of r.v.s

If
and
are jointly Gaussian random variables, then they
are independent iff
. (Show this using the p.d.f.)

Caution:
Gaussian r.v.s are special this way. As a general
rule,
uncorrelated does not imply independence.
.

In practice, it is common to assume that random variables are
independent based on physical arguments, rather than to prove is by
identifying a joint density and computing the marginals.

Many times, independence is also taken as an assumption, even when it
is not strictly true. This independence assumption frequently
simplifies analysis. However, the validity of the assumption must be
validated (e.g., using computer simulations).